Spectral properties of 1-D Schr\"odinger operators
$\mathrm{H}_{X,\alpha}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in
X}\alpha_n\delta(x-x_n)$ with local point interactions on a discrete set
$X=\{x_n\}_{n=1}^\infty$ are well studied when
$d_*:=\inf_{n,k\in\N}|x_n-x_k|>0$. Our paper is devoted to the case $d_*=0$. We
consider $\mathrm{H}_{X,\alpha}$ in the framework of extension theory of
symmetric operators by applying the technique of boundary triplets and the
corresponding Weyl functions.

We show that the spectral properties of $\mathrm{H}_{X,\alpha}$ like
self-adjointness, discreteness, and lower semiboundedness correlate with the
corresponding spectral properties of certain classes of Jacobi matrices. Based
on this connection, we obtain necessary and sufficient conditions for the
operators $\mathrm{H}_{X,\alpha}$ to be self-adjoint, lower-semibounded, and
discrete in the case $d_*=0$.

The operators with $\delta'$-type interactions are investigated too. The
obtained results demonstrate that in the case $d_*=0$, as distinguished from
the case $d_*>0$, the spectral properties of the operators with $\delta$ and
$\delta'$-type interactions are substantially different.